Why don't we perform classification of crowd density?
For example, if I have the crowd images labeled based on their crowd density (low, moderate, high), and I'm not interested in the count, but the density class (low, moderate, high), can't I train the network to classify the data based on these classes as a classification network?


Yes you can, all you need is enough correctly labelled training data.

A good rule of thumb is if a human expert can assign the correct label from an image (and purely from the image, not using extra information) then it is a realistic goal to train a CNN to perform the same labelling. 


  why there hasn't been attempts to tackle this type of task as a classification problem.


Probably because there are no natural, and likely no widely accepted, classes in this case (I may be wrong, maybe some international society has defined classes you could use). If you use a regression, you can map it to a particular problem case - e.g. sending an alert to someone responsible for traffic and safety when crowd density hits some threshold - by setting numerical boundaries to your classes. Using classification and mapping back the other way is harder.